L11a302

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_4,9,5,10 X_16,6,17,5 X_22,14,9,13 X_20,16,21,15 X_14,22,15,21 X_18,8,19,7 X_6,18,7,17 X_8,20,1,19
Gauss code	{1, -2, 3, -4, 5, -10, 9, -11}, {4, -1, 2, -3, 6, -8, 7, -5, 10, -9, 11, -7, 8, -6}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2(u-1)(v-1)(uv+1)^{2}}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-7q^{9/2}+8q^{7/2}-{\frac {1}{q^{7/2}}}-10q^{5/2}+{\frac {2}{q^{5/2}}}+10q^{3/2}-{\frac {4}{q^{3/2}}}+q^{15/2}-2q^{13/2}+4q^{11/2}-9{\sqrt {q}}+{\frac {6}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}-z^{7}a^{-3}+az^{5}-5z^{5}a^{-1}-5z^{5}a^{-3}+z^{5}a^{-5}+4az^{3}-8z^{3}a^{-1}-8z^{3}a^{-3}+4z^{3}a^{-5}+4az-4za^{-1}-4za^{-3}+4za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-2z^{9}a^{-1}-4z^{9}a^{-3}-2z^{9}a^{-5}+z^{8}a^{-2}+2z^{8}a^{-4}-2z^{8}a^{-6}-3z^{8}-3az^{7}+4z^{7}a^{-1}+16z^{7}a^{-3}+7z^{7}a^{-5}-2z^{7}a^{-7}-2a^{2}z^{6}+2z^{6}a^{-2}-3z^{6}a^{-4}+5z^{6}a^{-6}-z^{6}a^{-8}+9z^{6}-a^{3}z^{5}+9az^{5}+z^{5}a^{-1}-30z^{5}a^{-3}-14z^{5}a^{-5}+7z^{5}a^{-7}+5a^{2}z^{4}-7z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}a^{-6}+4z^{4}a^{-8}-10z^{4}+3a^{3}z^{3}-9az^{3}-13z^{3}a^{-1}+23z^{3}a^{-3}+18z^{3}a^{-5}-6z^{3}a^{-7}-a^{2}z^{2}+2z^{2}a^{-4}+z^{2}a^{-6}-4z^{2}a^{-8}+2z^{2}-a^{3}z+5az+6za^{-1}-6za^{-3}-5za^{-5}+za^{-7}+1-az^{-1}-a^{-1}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

1

12

3

1

-2

10

4

1

3

8

4

3

-1

6

4

2

4

0

2

5

6

-1

0

3

6

3

-2

1

3

-2

-4

1

3

2

-6

1

-1

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{5}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$